Particle Info Node in a Different Object
I don't know a exacly soution for your question. But there is a way to do something like that... Instead of using the Particle Livetime you can use the heat Attribute, a Math noder and the ColorRamp.


Rename the Attribute "color" in "heat" (or add a blank Attribute node)
Add a Math node, change to multiply
Enter a colorramp
Connect the Colorramp with the color input of Volume Scatter and Volume Absortion.
Play with the value of the Multiply Node and the Positions inside the Colorramp
If you want thicker Smoke increase the Multiply Value of the density
also play with the temperature settings of the Domain, and Flow 


If you know a better way, let me know....